https://file.aiquickdraw.com/mathbot/file/17286122314656kojzvqd.jpg

Given the triangle in the image, we are asked to find the exact value of \(\csc(Q)\) in its simplest form. Let's analyze the triangle and find the necessary information.

### Step 1: Understanding the Triangle
The triangle has vertices labeled as \(P\), \(Q\), and \(O\), with:
- \(PQ = 9\)
- \(OQ = \sqrt{85}\)
- \(PO = 2\)
- Angle \(P\) is marked as a right angle.

Since this is a right triangle, we can use trigonometric relationships to find \(\csc(Q)\).

### Step 2: Definition of \(\csc(Q)\)
The cosecant of an angle \(Q\) in a right triangle is defined as:
\[
\csc(Q) = \frac{\text{Hypotenuse}}{\text{Opposite side}}
\]
In this triangle, relative to angle \(Q\):
- The hypotenuse is \(OQ = \sqrt{85}\).
- The side opposite to angle \(Q\) is \(PQ = 9\).

### Step 3: Calculating \(\csc(Q)\)
\[
\csc(Q) = \frac{OQ}{PQ} = \frac{\sqrt{85}}{9}
\]

### Conclusion
The exact value of \(\csc(Q)\) in simplest form is:
\[
\csc(Q) = \frac{\sqrt{85}}{9}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on the topic:
1. How do you find the sine of angle \(Q\) using the same triangle?
2. What is the cosine of angle \(Q\) in this triangle?
3. How do you find the length of the adjacent side to angle \(Q\) using trigonometric ratios?
4. What is the value of \(\sec(P)\) in this right triangle?
5. How can you determine the area of this triangle using its side lengths?

**Tip:** In a right triangle, the trigonometric ratios of angles can often be quickly calculated using the lengths of the sides, making it easier to solve for other values when needed.

Find the exact value of csc Q in simplest form.

This problem involves finding the exact value of csc(Q) in a right triangle. Given side lengths, we calculate csc(Q) using trigonometric ratios and the Pythagoras theorem. Suitable for students in Grades 9-12.

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